SMO 2019

In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$ , points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$ . Prove that $\angle DMR=\angle MDR$ .

A robot is placed at point $P$ on the $x$ -axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$ . Player $B$ 's aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$ . For any integer $n$ , let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$ . At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$ . Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$ .

Proposed by fattypiggy123

In a $m\times n$ chessboard ( $m,n\ge 2$ ), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.

Proposed by DVDthe1st, mzy and jjax